7. In early 2018 TD Ameritrade and The Goldman Sachs Group , Inc. sold a bond with the following characteristics: . Coupon : 2.50 % to 03/29/2019 ; 3.50 % to 03/29/2020 ; 4.00 % to 03/29/2021 . 0 Payment Frequency: SemiAnnually. o Maturity: 03/29/2021. Derive a closed form solution for the price of this bond on 03/29/2018 using three annuities and one zero coupon bond . Solution : A step up bond can be constructed by summing annuities and adding a zerocoupon bond representing the principal payment at the end. To simplify the notation let's denote the coupon in years one, two and three by 01, 02, and 03, respectively. Also, recall that our general form for at-year annuity paying ntimes per year is P _C 1 amn(t)_ 71* W (14) The contribution from the rst year's coupons is 01 1 P1 = _ 1, , (15) Y (1+ Y/2l2 the contribution from the second year's coupons is an annuity that begins at the end of year one 02 1 1 P2 = _ 1 e 2 2 , (16) Y (1 + Y/2) (1 + Y/z) the contribution from the third year's coupons is an annuity that starts at the end of year two 03 1 1 p3=_1,_2_4_, (17) Y (1 + Y/zl (1 + '72) and the contribution from the final principal payment is 100 P4: _. (18) (1+ '72)" Thus , the price of the bond as a percent of par is P=P1+P2+P3+P4 (19)